Riemann–von Mangoldt formula explained

In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, describes the distribution of the zeros of the Riemann zeta function.

The formula states that the number N(T) of zeros of the zeta function with imaginary part greater than 0 and less than or equal to T satisfies

N(T) = (T)/(log { 2 \pi) (T)/(2 \pi) }-\frac+O(\log).

The formula was stated by Riemann in his notable paper "On the Number of Primes Less Than a Given Magnitude" (1859) and was finally proved by Mangoldt in 1905.

Backlund gives an explicit form of the error for all T > 2:

\left\vert{ N(T) - \left({ (T)/(log { 2 \pi) (T)/(2 \pi) }-\frac } - \frac\right)}\right\vert < 0.137 \log T + 0.443 \log\log T + 4.350 \ . Under the Lindelöf and Riemann hypotheses the error term can be improved to o( log {T}) and O( log {T}/ log { log {T}}) respectively.

Similarly, for any primitive Dirichlet character χ modulo q, we have

N(T,\chi) = (T)/(log { \pi) (qT)/(2 \pi e) }+O(\log),

where N(T,χ) denotes the number of zeros of L(s,χ) with imaginary part between -T and T.

References

• Book: Edwards, H.M. . Harold Edwards (mathematician)

. Harold Edwards (mathematician) . Riemann's zeta function . Pure and Applied Mathematics . 58 . New York-London . Academic Press . 1974 . 0-12-232750-0 . 0315.10035 .

• Book: Ivić, Aleksandar . The theory of Hardy's Z-function . Cambridge Tracts in Mathematics . 196 . Cambridge . . 2013 . 978-1-107-02883-8 . 1269.11075.
• Book: Patterson, S.J. . An introduction to the theory of the Riemann zeta-function . Cambridge Studies in Advanced Mathematics . 14 . Cambridge . . 1988 . 0-521-33535-3 . 0641.10029 .